The Acme Corporation has a total of 33 employees, each of whom works in one of three departments and at...

Phase 1: Owning the Dataset

Let's create a visualization to understand the employee distribution:

Total Employees: 33
├── Central Office: ? employees
│   ├── Department 1: some employees
│   ├── Department 2: some employees
│   └── Department 3: some employees
└── South Office: ? employees
    ├── Department 1: some employees
    ├── Department 2: some employees
    └── Department 3: some employees

Key facts:

• \(\mathrm{Total} = 33\) employees
• 3 departments
• 2 offices
• Each department has employees at BOTH offices

Phase 2: Understanding the Question

The statement says: "To determine the mean number of employees per department at South Office, first take the total number of employees in Central Office and A, and then take the resulting number and B."

Let's break this down:

• We want: Mean employees per department at South Office
• \(\mathrm{Mean} = \frac{\mathrm{Total\ employees\ at\ South\ Office}}{\mathrm{Number\ of\ departments}}\)
• Since there are 3 departments: \(\mathrm{Mean} = \frac{\mathrm{Total\ at\ South}}{3}\)

Now, what's the relationship between Central and South offices?

• \(\mathrm{Total\ at\ Central} + \mathrm{Total\ at\ South} = 33\)
• Therefore: \(\mathrm{Total\ at\ South} = 33 - \mathrm{Total\ at\ Central}\)

So our target calculation is:

\(\mathrm{Mean} = \frac{33 - \mathrm{Total\ at\ Central}}{3}\)

Phase 3: Finding the Answer

Let's trace through the operations:

Starting point: Total employees at Central Office

Operation A: We need to transform "Total at Central" into "Total at South"

• \(\mathrm{Total\ at\ South} = 33 - \mathrm{Total\ at\ Central}\)
• This means we need to subtract it from 33

Operation B: We need to get the mean per department

• \(\mathrm{Mean} = \frac{\mathrm{Total\ at\ South}}{3}\)
• This means we need to divide it by 3

Let's verify our logic:

1. Start with: Total at Central Office
2. Subtract it from 33 → gives us Total at South Office
3. Divide by 3 → gives us mean per department at South Office ✓

Phase 4: Solution

For A: subtract it from 33

For B: divide it by 3

This correctly calculates the mean number of employees per department at South Office.